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In statistics, degrees of freedom (df) play a crucial role in various calculations and are pivotal when dealing with the t-distribution. Degrees of freedom are essentially the number of values in a calculation that are free to vary. When you have a small sample size, like in our example with nine observations, it's important to consider how many values remain free after imposing some constraints, such as calculating a mean.
For a dataset with sample size () observations, the formula for computing degrees of freedom in the context of the t-distribution is straightforward:\[ df = n - 1 \]This formula subtracts one from the total number of samples. In our scenario, with 9 observations, the degrees of freedom equals 8. This figure influences the accuracy of the estimation of the population parameters. Understanding df helps to appropriately reference the correct row in statistical tables, crucial for determining other key values like the critical value in a t-distribution.

A confidence interval gives us a range of values within which we expect the true population parameter will fall. In this case, we are interested in finding a range where the true population mean (\mu) may lie with a certain level of confidence, based on our sample.
When constructing a confidence interval, the sample mean (\bar{x}), the standard deviation of the sample, and the corresponding critical value are critical components. For an 80% confidence interval, the interpretation is that if we were to take many samples and construct an interval from each of them, 80% of those intervals would contain the true population mean. This doesn't mean there is an 80% probability that the interval calculated from a single sample contains the mean, but rather it is a reflection of the method's accuracy over repeated sampling.
The formula for constructing a confidence interval for the population mean is:\[\text{Confidence Interval} = \bar{x} \pm \left( t\times \frac{s}{\sqrt{n}} \right)\]where \(t\) is the critical value from the t-distribution, \(s\) is the sample standard deviation, and \(n\) is the sample size.

The critical value is a threshold value that determines the boundary in hypothesis testing or the extent of a confidence interval. For constructing a confidence interval using a t-distribution, identifying the correct critical value is vital.
Critical values are derived from the probability of the sample mean differing from the population mean due to sampling variability. For an 80% confidence level, the remaining 20% is considered the risk or significance level (\alpha), which is split between both tails of the distribution. Thus, each tail gets an \alpha/2 of 10% in this exercise.
To find the critical value, one must look up the cumulative probability (or percentage point) in a t-distribution table that corresponds to:\[ Cumulative Probability = 1 - \frac{\alpha}{2} = 0.9 \]Given 8 degrees of freedom, you can locate the row with df = 8 and find the column for a cumulative probability of 0.9. As per the solution, the critical value we would use is approximately 1.397. This critical value helps determine the confidence interval width, influencing precision and reliability in estimating the population mean.